Comparison semigroups and algebras of transformations. (Q711608)

A set equipped with a certain quaternary comparison operation \((-,-)[-,-]\) is called a `comparison algebra'. A semigroup \((A,\cdot)\) equipped with a comparison operation \((-,-)[-,-]\) which satisfies the equalities \((a,b)[c,d]\cdot e=(a,b)[ce,de]\) and \(e\cdot(a,b)[c,d]=(ea,eb)[ec,ed]\) for all \(a,b,c,d,e\in A\), is called a `comparison semigroup'. A comparison semigroup embeddable in one of the form \(T_X\) (the semigroup of transformations on a set \(X\)) is called `functional'. It is proved that the abstract class of functional comparison semigroups (monoids) is the class of comparison semigroups (resp. monoids). Moreover, using a generalised comparison operation, the latter result is generalised to subalgebras of \(P_X\) (partial transformations on a set \(X\)). Finally, the author expresses a number of operations on partial transformations in the language of comparison semigroups with zero.